Mathematical and numerical challenges in diffuse optical tomography inverse problems

Computed Tomography (CT) is an essential imaging tool for medical inspection, diagnosis and prevention. While X-rays CT is a consolidated technology, there is nowadays a strong drive for innovation in this field. Between the emerging topics, Diffuse Optical Tomography (DOT) is an instance of Diffuse Optical Imaging which uses non-ionizing light in the near-infrared (NIR) band as investigating signal. Non-trivial challenges accompany DOT reconstruction, which is a severely ill-conditioned inverse problem due to the highly scattering nature of the propagation of light in biological tissues. Correspondingly, the solution of this problem is far from being trivial. In this review paper, we first recall the theoretical basis of NIR light propagation, the relevant mathematical models with their derivation in the perspective of a hierarchy of modeling approaches and the analytical results on the uniqueness issue and stability estimates. Then we describe the state-of-the-art in analytic theory and in computational and algorithmic methods. We present a survey of the few contributions regarding DOT reconstruction aided by machine learning approaches and we conclude providing perspectives in the mathematical treatment of this highly challenging problem.